0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to a quasi-smooth intersection subvariety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f s ⊂ P d Σ such that d + s = 2 ( k + 1 ) . If the Hodge conjecture holds on X then it holds as well on Y .\n\nCorollary 5.4. If the dimension of Y is 2 s − 1 , 2 s or 2 s + 1 then the Hodge conjecture holds on Y .\n\nProof. By Proposition 5.3 and Corollary 3.6.\n\n[\n\n] Angella, D. Cohomologies of certain orbifolds. Journal of Geometry and Physics\n\n(\n\n),\n\n–\n\n[\n\n] Batyrev, V. V., and Cox, D. A. On the Hodge structure of projective hypersur- faces in toric varieties. Duke Mathematical Journal\n\n,\n\n(Aug\n\n). [\n\n] Bruzzo, U., and Montoya, W. On the Hodge